On supra soft topological ordered spaces

Abstract

In this work, we initiate the concept of supra soft topological ordered spaces which are consider as an extension of the concept of soft topological ordered spaces. We define and discuss some notions via supra soft topological ordered spaces such as monotone interior, closure and limit operators. Then, we formulate some supra soft separation axioms, namely supra p-soft T_i-ordered spaces $(i=0,1,2,3,4).$ These axioms are studied in terms of the ordinary points and the two relations of natural belong and total non-belong. We provide some illustrative examples to show the relationships between them and to investigate the mutual relations between them and their parametric supra topologies. Additionally, we characterize the concepts of supra p-soft T_i-ordered spaces (i = 1, 2), supra p-soft regularly ordered and supra soft normally ordered spaces. Finally, we conclude some findings related to hereditary and topological properties and finite product spaces.

Keywords:

• Supra soft topological ordered space
• supra p-soft Ti-ordered space
• finite product spaces

2010 Mathematics Subject Classification:

• 54A05
• 54B10
• 54D10
• 54D15
• 54F05

1. Introduction

There are many mathematical tools available for modelling complex systems or for dealing with imperfect knowledge such as probability theory, fuzzy set theory, rough set theory, etc. But there are inherent difficulties associated with each of these techniques. All these tools require the pre-specification of some parameters to start with, for example, probability density function in probability theory, membership function in fuzzy set theory and an equivalence relation in rough set theory. Such a requirement, seen in the backdrop of imperfect or incomplete knowledge, raises many problems. To overcome these obstacles, Molodtsov (1999) proposed a new mathematical tool, namely soft sets. It is more virtual than the previous tools for dealing with uncertainties or incomplete data. The interested readers can refer to Molodtsov’s work to know its metrics and potential applications in several directions.

Shabir and Naz (2011) defined the concept of soft topologies based on soft sets. They fixed a parameters set to remove shortcoming of non-existence uniquely null soft set, and they utilized a relative complement of a soft set to define soft closed sets in order to keep some set theoretic properties via soft topology. Later on, many authors carried out detailed studies via soft topologies and defined most of the fundamental concepts in soft topologies analogously to them in ordinary topologies. In particular, they highlighted on soft separation axioms which are among the most interesting and common topics via soft topologies (see, e.g., Al-shami & El-Shafei, 2019a; Bayramov & Aras, 2018; El-Shafei, Abo-Elhamayel, & Al-shami, 2018; Singh & Noorie, 2017; Tantawy, El-Sheikh, & Hamde, 2016). Al-shami and Kočinac (2019) proved the equivalence between enriched and extended soft topologies and derived the interchangeable attribute for soft interior and closure operators.

El-Sheikh and Abd El-Latif (2014) relaxed the conditions of a soft topology to construct a wider and more general class, namely a supra soft topology. It is expected that many considerable results in soft topologies will not be carried over and some of interesting properties will be missing or weakened in supra soft topologies. However, in order to attain desirable and interesting conclusions, additional conditions must be imposed. Abo-Elhamayel and Al-shami (2016) studied supra soft axioms via supra soft topological spaces with respect to the soft points. Al-shami, El-Shafei, and Abo-Elhamayel (2018a) constructed the concept of soft topological ordered spaces and studied some soft ordered axioms, namely p-soft T_i-ordered spaces $(i=0,1,2,3,4).$ Then they (Al-shami, El-Shafei, & Abo-Elhamayel, 2018b) defined and studied soft ordered maps between soft topological ordered spaces. El-Shafei, Abo-Elhamayel, and Al-shami (2019) investigated more notions via supra soft topologies such as supra soft compactness and supra soft closure operator. Recently, Al-shami and El-Shafei (2019b) studied two new types of soft axioms via supra soft topological spaces with respect to the ordinary points, whereas Aras and Bayramov (2019) explored new soft axioms via supra soft topological spaces with respect to the soft points. Supra soft topological spaces were generalized in many ways (see, e.g., Aras, 2018; Aras & Bayramov, 2018).

The present work aims to define a new soft structure, namely supra soft topological spaces and to study some notions via this structure. We introduce new types of supra soft separation axioms with respect to the ordinary points, namely supra p-soft T_i-ordered spaces $(i=0,1,2,3,4).$ One of the motivations to study these axioms is to establish a wider family which can be easily applied to classify the objects of study and to show the significant role of a total non-belong relation in obtaining results similar to those via supra topologies. In general, we probe the interrelations between these axioms and some concepts such as parametric supra topologies, hereditary and topological properties and finite product spaces.

2. Preliminaries

We allocate this section to recall some definitions and results which we need them in the material of this work.

Definition 2.1.

(Molodtsov, 1999) A pair (G, T) is said to be a soft set over a non-empty set A provided that G is a mapping of a set of parameters T into $2^A.$ It can be written (G, T) as a set of ordered pairs: $(G,T)=\{(t,G(t)):t\in T$ and $G(t)\in 2^A\}.$

Remark 2.2.

1. The collection of all soft sets over A under a parameters set T is denoted by $Z(A_T).$

2. We sometimes write $F,H,U,V$ in a place of G, M, N in a place of T, and B in a place of A.

Definition 2.3.

(Ali et al., 2009) The relative complement of a soft set (G, T), denoted by ${(G,T)}^c,$ is given by ${(G,T)}^c=(G^c,T),$ where $G^c:T\to 2^A$ is a mapping defined by $G^c(t)=A\setminus G(t)$ for each $t\in T.$

Definition 2.4.

(Maji, Biswas, & Roy, 2003) A soft set (G, T) over A which satisfies that $G(t)=\varnothing$ for each $t\in T$ is said to be a null soft set and its relative complement is said to be an absolute soft set. The null and absolute soft sets are denoted respectively by $\tilde{\Phi }$ and $\tilde{A}.$

Definition 2.5.

(Feng, Li, Davvaz, & Ali, 2010) A soft set (G, M) is a subset of a soft set (F, N), denoted by $(G,M)\tilde{\subseteq }(F,N),$ if $M\subseteq N$ and for all $m\in M,$ we have $G(m)\subseteq F(m).$

The soft sets (G, M) and (F, N) are soft equal if each one of them is a soft subset of the other.

Definition 2.6.

(Maji et al., 2003) The union of two soft sets (G, M) and (F, N) over A, denoted by $(G,M)\tilde{\cup }(F,N),$ is a soft set (H, S), where $S=M\cup N$ and a mapping $H:S\to 2^A$ is given as follows: $$H(s)=\{\begin{array}{lll}G(s)& :& s\in M\setminus N\\ F(s)& :& s\in N\setminus M\\ G(s)\cup F(s)& :& s\in M\cap N\end{array}$$

Definition 2.7.

(Ali et al., 2009) The intersection of two soft sets (G, M) and (F, N) over A, denoted by $(G,M)\tilde{\cap }(F,N),$ is a soft set (H, S), where $S=M\cap N\ne \varnothing$ and a mapping $H:S\to 2^A$ is given by $H(s)=G(s)\cap F(s).$

Definition 2.8.

(Das & Samanta, 2013; Nazmul & Samanta, 2013) A soft set (P, T) over A is called soft point if there exists $t\in T$ and there exists $a\in A$ such that $P(t)=\left\{a\right\}$ and $P(\alpha )=\varnothing$ for each $\alpha \in T\setminus \left\{t\right\}.$ A soft point will be shortly denoted by $P_t^a.$ We write $P_t^a\in (G,T)$ if $a\in G(t).$

Definition 2.9.

(Aygünoǧlu & Aygün, 2012) Let (G, M) and (F, N) be two soft sets over A and B, respectively. The cartesian product of (G, M) and (F, N), denoted by $(G\times F,M\times N),$ is defined by $(G\times F)(m,n)=G(m)\times F(n)$ for each $(m,n)\in M\times N.$

Definition 2.10.

(Zorlutuna, Akdag, Min, & Atmaca, 2012) A soft mapping between $Z(A_T)$ and $Z(B_S)$ is a pair $(f,\varphi ),$ denoted also by $f_{\varphi },$ of the mappings $f:A\to B,\varphi :T\to S.$ Let (G, M) and (F, N) be soft subsets of $Z(A_T)$ and $Z(B_S),$ respectively. Then the image of (G, M) and pre-image of (F, N) are given by:

1. $f_{\varphi }(G,M)={(f_{\varphi }(G))}_S$ is a soft subset of $Z(B_S)$ such that $$f_{\varphi }(G)(s)=\{\begin{array}{lll}\underset{m\in {\varphi }^{-1}(s)\cap M}{\cup }f(G(m))& :& {\varphi }^{-1}(s)\cap M\ne \varnothing \\ \varnothing & :& {\varphi }^{-1}(s)\cap M=\varnothing \end{array}$$ for each $s\in S.$

2. $f_{\varphi }^{-1}(F,N)={(f_{\varphi }^{-1}(F))}_T$ is a soft subset of $Z(A_T)$ such that $$f_{\varphi }^{-1}(F)(t)=\{\begin{array}{lll}{f}^{-1}(F(\varphi (t)))& :& \varphi (t)\in N\\ \varnothing & :& \varphi (t)\notin N\end{array}$$ for each $t\in T.$

Definition 2.11.

(Zorlutuna et al., 2012) A soft mapping $f_{\varphi }:Z(A_T)\to Z(B_S)$ is said to be injective (resp. surjective, bijective) provided that f and $\varphi$ are injective (resp. surjective, bijective).

Definition 2.12.

(El-Shafei et al., 2018; Shabir & Naz, 2011) Let (G, T) be a soft set over A and let $a\in A.$ We write:

1. $a⋐(G,T)$ if $a\in G(t)$ for some $t\in T;$ and $a\cancel{⋐}(G,T)$ if $a\notin G(t)$ for each $t\in T.$

2. $a\in (G,T)$ if $a\in G(t)$ for each $t\in T;$ and $a\notin (G,T)$ if $a\notin G(t)$ for some $t\in T.$

Definition 2.13.

A binary relation $⪯$ on a non-empty set is called partial order relation if it is reflexive, anti-symmetric and transitive. In particular, the diagonal relation on any non-empty set shall be shortly denoted by $△$ and the usual partial order relation on the set of integer numbers Z is defined as follows $⪯=\{(a,b):a\le b$ for each $a,b\in \mathbf{Z}\}.$

Definition 2.14.

Let $(A,⪯)$ be a partially ordered set. An element $b\in A$ is called:

1. A smallest element of A provided that $b⪯a$ for all $a\in A.$

2. A largest element of A provided that $a⪯b$ for all $a\in A.$

Definition 2.15.

(Al-shami et al. 2018a) Let $⪯$ be a partial order relation on a non-empty set A and let T be a set of parameters. A triple $(A,T,⪯)$ is said to be a partially ordered soft set. For two soft points $P_{\alpha }^a$ and $P_{\alpha }^b,$ we write $P_{\alpha }^a⪯P_{\alpha }^b\iff a⪯b.$

Definition 2.16.

(Al-shami et al. 2018a) A soft mapping $f_{\varphi }:(Z(A_T),{⪯}_1)\to (Z(B_S),{⪯}_2)$ is said to be ordered embedding provided that $P_{\alpha }^a{⪯}_1{P}_{\alpha }^b$ if and only if $f_{\varphi }(P_{\alpha }^a){⪯}_2{f}_{\varphi }(P_{\alpha }^b).$

Definition 2.17.

(Al-shami et al. 2018a) We define an increasing soft operator $i:(Z(A_T),⪯)\to (Z(A_T),⪯)$ and a decreasing soft operator $d:(Z(A_T),⪯)\to (Z(A_T),⪯)$ as follows: For each soft subset (G, T) of $Z(A_T)$

1. $i(G,T)=(\mathit{iG},T),$ where iG is a mapping of T into A given by $\mathit{iG}(t)=i(G(t))=\{a\in A:b⪯a$ for some $b\in G(t)\}.$

2. $d(G,T)=(\mathit{dG},T),$ where dG is a mapping of T into A given by $\mathit{dG}(t)=d(G(t))=\{a\in A:a⪯b$ for some $b\in G(t)\}.$

Definition 2.18.

(Al-shami et al. 2018a) A soft subset (G, T) of a partially ordered soft set $(A,T,⪯)$ is said to be increasing (resp. decreasing) if $(G,T)=i(G,T)($resp. $(G,T)=d(G,T)).$

Now, we give the following example to show that the three definitions above.

Example 2.19.

Let $A=\{a_1,a_2\},B=\{b_1,b_2\},T=\{t_1,t_2\}$ and $S=\{s_1,s_2\}.$ Consider ${⪯}_1=△\cup \{(a_1,a_2)\}$ and ${⪯}_2=△\cup \{(b_1,b_2)\}$ be two partial order relations on A and B, respectively. Then $(A,T,{⪯}_1)$ and $(B,S,{⪯}_2)$ are partially ordered soft sets. A soft subset $(G,T)=\{(t_1,\left\{a_1\right\}),(t_2,\left\{a_2\right\})\}$ of $(Z(A_T),{⪯}_1)$ is neither increasing nor decreasing because $i(G,T)=\{(t_1,X),(t_2,\left\{a_2\right\})\}\ne (G,T)$ and $d(G,T)=\{(t_1,\left\{a_1\right\}),(t_2,X)\}\ne (G,T).$ On the other hand, the soft subsets $(F,T)=\{(t_1,\left\{a_2\right\}),(t_2,\left\{a_2\right\})\}$ and $(G,T)=\{(t_1,\varnothing ),(t_2,\left\{a_1\right\})\}$ of $(Z(A_T),{⪯}_1)$ are increasing and decreasing, respectively.

Consider a soft mapping $f_{\varphi }:(Z(A_T),{⪯}_1)\to (Z(B_S),{⪯}_2),$ where the maps $f:A\to B$ and $\varphi :T\to S$ defined as follows $f(a_1)=b_1,f(a_2)=b_2$ and $\varphi (t_1)=\varphi (t_2)=s_1.$ Now, $f_{\varphi }(P_{t_1}^{a_1})=f_{\varphi }(P_{t_2}^{a_1})=P_{s_1}^{b_1},$ and $f_{\varphi }(P_{t_1}^{a_2})=f_{\varphi }(P_{t_2}^{a_2})=P_{s_1}^{b_2}.$ Since $a_1{⪯}_1{a}_2,$ then $P_{t_1}^{a_1}{⪯}_1{P}_{t_1}^{a_2}$ and $P_{t_2}^{a_1}{⪯}_1{P}_{t_2}^{a_2};$ and since $b_1{⪯}_2{b}_2,$ then $P_{s_1}^{b_1}{⪯}_2{P}_{s_1}^{b_2}$ and $P_{s_2}^{b_1}{⪯}_2{P}_{s_2}^{b_2}.$ This means that $P_{t_1}^{a_1}{⪯}_1{P}_{t_1}^{a_2}$ if and only if $f_{\varphi }(P_{t_1}^{a_1}){⪯}_2{f}_{\varphi }(P_{t_1}^{a_2});$ and $P_{t_2}^{a_1}{⪯}_1{P}_{t_2}^{a_2}$ if and only if $f_{\varphi }(P_{t_2}^{a_1}){⪯}_2{f}_{\varphi }(P_{t_2}^{a_2}).$ Hence, $f_{\varphi }$ is ordered embedding.

Theorem 2.20.

(Al-shami et al. 2018a) The finite product of increasing (resp. decreasing) soft sets is increasing (resp. decreasing).

Proposition 2.21.

(Al-shami et al. 2018a) Let $f_{\varphi }:Z(A_T)\to Z(B_S)$ be a soft mapping and let (G, M) and (F, N) be soft sets in $Z(A_T)$ and $Z(B_S)$, respectively. Then the following statements hold.

1. If f is injective and $a\cancel{⋐}(G,M)$, then $f(a)\cancel{⋐}{f}_{\varphi }(G,M).$

2. If $\varphi$ is surjective and $a\in (G,M)$, then $f(a)\in f_{\varphi }(G,M).$

3. If $f_{\varphi }$ is injective and $a\notin (G,M)$, then $f(a)\notin f_{\varphi }(G,M).$

4. If $b\cancel{⋐}(F,N)$, then $a\cancel{⋐}{f}_{\varphi }^{-1}(F,N)$ for each $a\in f^{-1}(b).$

5. If $b\in (F,N)$, then $a\in f_{\varphi }^{-1}(F,N)$ for each $a\in f^{-1}(b).$

6. If $\varphi$ is surjective and $b\notin (F,N)$, then $a\notin f_{\varphi }^{-1}(F,N)$ for each $a\in f^{-1}(b).$

Theorem 2.22.

Let $f_{\varphi }:(Z(A_T),{⪯}_1)\to (Z(B_S),{⪯}_2)$ be a bijective ordered embedding soft map. Then the image of each increasing (resp. decreasing) soft subset of $\tilde{A}$ is an increasing (resp. a decreasing) soft subset of $\tilde{B}.$

Definition 2.23.

(Shabir & Naz, 2011) A soft set (a, T) over A is defined by $a(t)=\left\{a\right\}$ for each $t\in T.$

Definition 2.24.

(El-Sheikh & Abd El-Latif, 2014) The collection μ of soft sets over A under a fixed parameters set T is said to be a supra soft topology on A if the following two axioms hold:

1. $\tilde{A}$ and $\tilde{\Phi }$ belong to μ.

2. The union of an arbitrary family of soft sets in μ belongs to μ.

The triple $(A,\mu ,T)$ is called a supra soft topological space. The members of μ and their relative complement are called respectively supra soft open sets and supra soft closed sets.

Definition 2.25.

(El-Sheikh & Abd El-Latif, 2014) For a soft subset (H, T) of $(A,\mu ,T),$ we define the following:

1. ${(H,T)}^{\mathit{so}}$ is the union of all supra soft open sets contained in (H, T).

2. ${(H,T)}^{\mathit{scl}}$ is the intersection of all supra soft closed sets containing (H, T).

Theorem 2.26.

(El-Shafei et al., 2019) Let (H, T) and (F, T) be two soft subsets of $(A,\mu ,T)$. Then:

1. If $(H,T)\tilde{\subseteq }(F,T)$, then ${(H,T)}^{\mathit{scl}}\tilde{\subseteq }{(F,T)}^{\mathit{scl}}.$

2. $P_t^a\in {(H,T)}^{\mathit{scl}}$ if and only if $(G,T)\tilde{\cap }(H,T)\ne \tilde{\Phi }$ for each supra soft open set (G, T) containing $P_t^a.$

Definition 2.27.

(El-Shafei et al., 2019) Let $(A,\mu ,T)$ be a supra soft topological space and Y be a non-empty subset of A. Then ${\mu }_Y=\{\tilde{Y}\tilde{\cap }(G,T):(G,T)\in \mu \}$ is called a relative supra soft topology on Y and $(Y,{\mu }_Y,T)$ is called a supra soft subspace of $(A,\mu ,T).$

Theorem 2.28.

(El-Shafei et al., 2019) Let $(Y,{\mu }_Y,T)$ be a supra soft subspace of $(A,\mu ,T)$. Then (H, T) is a supra soft closed subset of $(Y,{\mu }_Y,T)$ if and only if there exists a supra soft closed subset (F, T) of $(A,\mu ,T)$ such that $(H,T)=(F,T)\tilde{\cap }\tilde{Y}.$

Definition 2.29.

(Abd El-Latif & Hosny, 2017) A supra soft topological space $(A,\mu ,T)$ is called supra soft normal if for every two disjoint supra soft closed sets $(H_1,T)$ and $(H_2,T),$ there exist two disjoint supra soft open sets $(G_1,T)$ and $(G_2,T)$ such that $(H_1,T)\tilde{\subseteq }(G_1,T)$ and $(H_2,T)\tilde{\subseteq }(G_2,T).$

Definition 2.30.

(Al-shami & El-Shafei, 2019b) Let $\{(A_i,{\mu }_i,T_i):i=1,2,\dots ,n\}$ be the collection of supra soft topological spaces. Then ${\prod }_{i=1}^n{\mu }_i=\{{\prod }_{i=1}^n(G,T_i):(G,T_i)\in {\mu }_i\}$ defines a supra soft topology on ${\prod }_{i=1}^n{A}_i$ under a parameters set ${\prod }_{i=1}^n{T}_i.$ We call ${\prod }_{i=1}^n{\mu }_i$ a finite product supra soft topology and $({\prod }_{i=1}^n{A}_i,{\prod }_{i=1}^n{\mu }_i,{\prod }_{i=1}^n{T}_i)$ a finite product supra soft space.

Lemma 2.31.

(Al-shami & El-Shafei, 2019b) If $(H,T_1\times T_2)$ is a supra soft closed subset of a product supra soft space $(A\times B,{\mu }_1\times {\mu }_2,T_1\times T_2)$, then $(H,T_1\times T_2)=[{(G,T_1)}^c\times \tilde{B}]\tilde{\cup }[\tilde{A}\times {(U,T_2)}^c]$ for some $(G,T_1)\in {\mu }_1$ and $(U,T_2)\in {\mu }_2.$

Lemma 2.32.

(Al-shami & El-Shafei, 2019b) If (U, T) is an increasing (resp. a decreasing) soft subset of a partially ordered soft set $(A,T,⪯)$, then $(U,T)\tilde{\cap }\tilde{Y}$ is an increasing (resp. a decreasing) soft subset of $(Y,T,{⪯}_Y).$

Definition 2.33.

(Al-shami & El-Shafei, 2019b) A soft mapping $f_{\varphi }:(A,\mu ,T)\to (B,\theta ,S)$ is called:

1. Soft $S^{\star }$-continuous if the inverse image of each supra soft open subset of $(B,\theta ,S)$ is a supra soft open subset of $(A,\mu ,T).$

2. Soft $S^{\star }$-open (resp. Soft $S^{\star }$-closed) if the image of each supra soft open (resp. supra soft closed) subset of $(A,\mu ,T)$ is a supra soft open (resp. supra soft closed) subset of $(B,\theta ,S).$

3. Soft $S^{\star }$-homeomorphism if it is bijective, soft $S^{\star }$-continuous and soft $S^{\star }$-open.

Definition 2.34.

(Das, 2004; Mashhour, Allam, Mahmoud, & Khedr, 1983) The sub-collection μ of $2^A$ is said to be a supra topology on a non-empty set A if it is closed under arbitrary union and contains the two sets A and $\varnothing .$ The triple $(A,\mu ,⪯)$ is said to be a supra topological ordered space, where $⪯$ is a partial order relation on A.

Some works on supra topological ordered spaces which investigated some types of ordered maps and ordered axioms can be found on (Abo-Elhamayel & Al-shami, 2016; El-Shafei, Abo-Elhamayel, & Al-shami, 2017).

3. Supra soft topological ordered spaces

We devote this section to introducing the concept of supra soft topological ordered spaces which is an extension of both soft topological ordered and supra soft topological spaces. Then we originate some notions via this concept and discuss their main features.

Definition 3.1.

A quadrable system $(A,\mu ,T,⪯)$ is said to be a supra soft topological ordered space, where $(A,\mu ,T)$ is a supra soft topological space and $(A,T,⪯)$ is a partially ordered soft set. Henceforth, we use the abbreviation SSTOS in a place of supra soft topological ordered space.

Proposition 3.2.

Let $f_{\varphi }:Z(A_T)\to Z(B_S)$ be a soft mapping such that f is injective. If $(B,\theta ,S,⪯)$ is a supra soft topological ordered space over B, then $(A,\mu ,T,\rho )$ is a supra soft topological ordered space over A, where $\mu =\{f_{\varphi }^{-1}(G,S):(G,S)\in \theta \}$ and $\rho =\{(a,b)\in A\times A:a\rho b\iff f(a)⪯f(b)\}.$

Proof.

First, we prove that μ is a supra soft topology over A. Owing to ${\tilde{\Phi }}_B$ and $\tilde{B}$ belong to θ, then $f_{\varphi }^{-1}({\tilde{\Phi }}_B)={\tilde{\Phi }}_A$ and $f_{\varphi }^{-1}(\tilde{B})=\tilde{A}$ belong to μ. Let $(F_i,T)\in \mu ,$ where $i\in I.$ Then $(F_i,T)=f_{\varphi }^{-1}(G_i,S),$ where $(G_i,S)\in \theta$ for each $i\in I.$ Now, ${\cup }_{i\in I}^{\sim }(F_i,T)={\cup }_{i\in I}^{\sim }{f}_{\varphi }^{-1}(G_i,S)=f_{\varphi }^{-1}({\cup }_{i\in I}^{\sim }(G_i,S)).$ Since ${\cup }_{{}_{i\in I}}^{\sim }(G_i,S)\in \theta ,$ then ${\cup }_{i\in I}^{\sim }(F_i,T)\in \mu .$ This shows that μ is closed under arbitrary soft union. Thus, μ is a supra soft topology over A.

Second, we prove that ρ is a partial order relation on A. For each $a\in A,$ we find that $f(a)⪯f(a).$ Then $a\rho a,$ so that ρ is reflexive. Let $a\rho b$ and $b\rho a.$ Then $f(a)⪯f(b)$ and $f(b)⪯f(a).$ Therefore, $f(a)=f(b).$ Since f is injective, then a = b. This means ρ is anti-symmetric. To show the transitivity of ρ, let $a\rho b$ and $b\rho c.$ Then $f(a)⪯f(b)$ and $f(b)⪯f(c).$ Therefore, $f(a)⪯f(c).$ By the definition of ρ, we obtain $a\rho c.$ This demonstrates that ρ is a partial order relation on A.

Hence, $(A,\mu ,T,\rho )$ is a supra soft topological ordered space over A. □

Proposition 3.3.

Let $f_{\varphi }:Z(A_T)\to Z(B_S)$ be a bijective soft mapping. If $(A,\mu ,T,\rho )$ is a supra soft topological ordered space over A, then $(B,\theta ,S,⪯)$ is a supra soft topological ordered space over B, where $\theta =\{f_{\varphi }(G,T):(G,T)\in \mu \}$ and $⪯=\{(x,y)\in B\times B:x\rho y\iff f^{-1}(x)⪯f^{-1}(y)\}.$

Proof.

The proof is similar to that of Proposition (3.2). □

Definition 3.4.

A soft subset (W, T) of an SSTOS $(A,\mu ,T,⪯)$ is said to be:

1. A supra soft neighbourhood of $a\in A$ if there exists a supra soft open set (G, T) such that $a\in (G,T)\tilde{\subseteq }(W,T).$

2. An increasing supra soft neighbourhood of $a\in A$ if (W, T) is a supra soft neighbourhood of a and increasing.

3. A decreasing supra soft neighbourhood of $a\in A$ if (W, T) is a supra soft neighbourhood of a and decreasing.

Definition 3.5.

For two soft subsets (G, T) and (H, T) of an SSTOS $(A,\mu ,T,⪯)$ and $a\in A,$ we say that:

1. (G, T) contains a provided that $a\in (G,T).$

2. (G, T) contains (H, T) provided that $(H,T)\tilde{\subseteq }(G,T).$

3. (G, T) is a supra soft neighbourhood of (H, T) provided that there exists a supra soft open set (F, T) such that $(H,T)\tilde{\subseteq }(F,T)\tilde{\subseteq }(G,T).$

Definition 3.6.

For a soft subset (H, T) of an SSTOS $(A,\mu ,T,⪯),$ we define the following operators:

1. ${(H,T)}^{\mathit{iso}}($resp. ${(H,T)}^{\mathit{dso}})$ is the union of all increasing (resp. decreasing) supra soft open set contained in $(H,T).$

2. ${(H,T)}^{\mathit{iscl}}($resp. ${(H,T)}^{\mathit{dscl}})$ is the intersection of all increasing (resp. decreasing) supra soft closed set containing $(H,T).$

Remark 3.7.

It can be noted the following:

1. ${(H,T)}^{\mathit{iso}}($resp. ${(H,T)}^{\mathit{dso}})$ is the largest increasing (resp. decreasing) supra soft open set contained in $(H,T).$

2. ${(H,T)}^{\mathit{iscl}}($resp. ${(H,T)}^{\mathit{dscl}})$ is the smallest increasing (resp. decreasing) supra soft closed set containing $(H,T).$

Proposition 3.8.

We have the following two properties for a soft subset (H, T) of an SSTOS $(A,\mu ,T,⪯)$:

1. ${[{(H,T)}^{\mathit{dscl}}]}^c={({(H,T)}^c)}^{\mathit{iso}}.$

2. ${[{(H,T)}^{\mathit{iscl}}]}^c={({(H,T)}^c)}^{\mathit{dso}}.$

Proof.

(i) ${[{(H,T)}^{\mathit{dscl}}]}^c=[\tilde{\cap }(F,T):(F,T)$ is a decreasing supra soft closed set containing $(H,T){]}^c$=$[\tilde{\cup }{(F,T)}^c:{(F,T)}^c$ is an increasing supra soft open set contained in ${(H,T)}^c]={({(H,T)}^c)}^{\mathit{iso}}.$

By analogy with (i), one can prove (ii). □

The following two propositions can be proven easily.

Proposition 3.9.

We have the following four properties for a soft subset (H, T) of an SSTOS $(A,\mu ,T,⪯)$:

1. (H, T) is increasing (resp. decreasing) supra soft open if and only if $(H,T)={(H,T)}^{\mathit{iso}}$ (resp. $(H,T)={(H,T)}^{\mathit{dso}}$).

2. (H, T) is increasing (resp. decreasing) supra soft closed if and only if $(H,T)={(H,T)}^{\mathit{iscl}}$ (resp. $(H,T)={(H,T)}^{\mathit{dscl}}$).

3. ${(H,T)}^{\mathit{iso}}\tilde{\subseteq }{(H,T)}^{\mathit{so}}$ and ${(H,T)}^{\mathit{dso}}\tilde{\subseteq }{(H,T)}^{\mathit{so}}.$

4. ${(H,T)}^{\mathit{scl}}\tilde{\subseteq }{(H,T)}^{\mathit{iscl}}$ and ${(H,T)}^{\mathit{scl}}\tilde{\subseteq }{(H,T)}^{\mathit{dscl}}.$

Proposition 3.10.

Let (H, T) and (G, T) be two soft subsets of an SSTOS $(A,\mu ,T,⪯)$ such that $(H,T)\tilde{\subseteq }(G,T)$. Then the following two properties hold.

1. ${(H,T)}^{\mathit{iso}}\tilde{\subseteq }{(G,T)}^{\mathit{iso}}$ and ${(H,T)}^{\mathit{dso}}\tilde{\subseteq }{(G,T)}^{\mathit{dso}}.$

2. ${(H,T)}^{\mathit{iscl}}\tilde{\subseteq }{(G,T)}^{\mathit{iscl}}$ and ${(H,T)}^{\mathit{dscl}}\tilde{\subseteq }{(G,T)}^{\mathit{dscl}}.$

Corollary 3.11.

Let (H, T) and (G, T) be two soft subsets of an SSTOS $(A,\mu ,T,⪯)$. Then the following four properties hold.

1. ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{iso}}\tilde{\subseteq }{(H,T)}^{\mathit{iso}}\tilde{\cap }{(G,T)}^{\mathit{iso}}$ and ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{dso}}\tilde{\subseteq }{(H,T)}^{\mathit{dso}}\tilde{\cap }{(G,T)}^{\mathit{dso}}.$

2. ${(H,T)}^{\mathit{iso}}\tilde{\cup }{(G,T)}^{\mathit{iso}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{iso}}$ and ${(H,T)}^{\mathit{dso}}\tilde{\cup }{(G,T)}^{\mathit{dso}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{dso}}.$

3. ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{iscl}}\tilde{\subseteq }{(H,T)}^{\mathit{iscl}}\tilde{\cap }{(G,T)}^{\mathit{iscl}}$ and ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{dscl}}\tilde{\subseteq }{(H,T)}^{\mathit{dscl}}\tilde{\cap }{(G,T)}^{\mathit{dscl}}.$

4. ${(H,T)}^{\mathit{iscl}}\tilde{\cup }{(G,T)}^{\mathit{iscl}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{iscl}}$ and ${(H,T)}^{\mathit{dscl}}\tilde{\cup }{(G,T)}^{\mathit{dscl}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{dscl}}.$

To see that the converse of the above results need not be true, we give the following example.

Example 3.12.

Let $T=\{t_1,t_2\}$ be a set of parameters and $⪯$ be the usual partial order relation on the set of integer numbers Z. Then $\mu =\{\tilde{\Phi },(G,T)\tilde{\subseteq }\tilde{\mathbf{Z}}$ such that $2\in (G,T)$ or $1⋐(G,T)\}$ is a supra soft topology on Z. Consider the following soft sets defined as follows: $$(H_1,T)=\{(t_1,\{1,3\}),(t_2,\{2,3\})\};$$ $$(H_2,T)=\{(t_1,\{1,2\}),(t_2,\{1,2\})\};$$ $$(H_3,T)=\{(t_1,\{1,2,\dots ,n,n+1,\dots \}),(t_2,\{\dots ,-n,-n+1,\dots ,1\})\};$$ $$(H_4,T)=\{(t_1,\{\dots ,-n,-n+1,\dots ,1\}),(t_2,\{1,2,\dots ,n,n+1,\dots \})\};$$

It can be seen that:

1. ${(H_1,T)}^{\mathit{so}}=(H_1,T),$ but ${(H_1,T)}^{\mathit{iso}}={(H_1,T)}^{\mathit{dso}}=\tilde{\Phi }.$

2. ${(H_1,T)}^{\mathit{scl}}=(H_1,T),$ but ${(H_1,T)}^{\mathit{iscl}}=\{(t_1,\{1,2,\dots ,n,n+1,\dots \}),(t_2,\{2,3,\dots ,n,n+1,\dots \})\}$ and ${(H_1,T)}^{\mathit{dscl}}=\tilde{\mathbf{Z}}.$

3. ${(H_3,T)}^{\mathit{iso}}=\{(t_1,\{1,2,\dots ,n,n+1,\dots \}),(t_2,\varnothing )\}$ and ${(H_3,T)}^{\mathit{dso}}=\{(t_1,\varnothing ),(t_2,\{\dots ,-n,-n+1,\dots ,1\})\}.$ Now, ${(H_1,T)}^{\mathit{iso}}\tilde{\subseteq }{(H_3,T)}^{\mathit{iso}}$ and ${(H_1,T)}^{\mathit{dso}}\tilde{\subseteq }{(H_3,T)}^{\mathit{dso}},$ but $(H_1,T)\tilde{\cancel{\subseteq }}(H_3,T).$

4. ${(H_2,T)}^{\mathit{iscl}}={(H_2,T)}^{\mathit{dscl}}=\tilde{\mathbf{Z}}.$ Now, ${(H_1,T)}^{\mathit{iscl}}\tilde{\subseteq }{(H_2,T)}^{\mathit{iscl}}$ and ${(H_1,T)}^{\mathit{dscl}}\tilde{\subseteq }{(H_2,T)}^{\mathit{dscl}},$ but $(H_1,T)\tilde{\cancel{\subseteq }}(H_2,T).$

5. ${(H_4,T)}^{\mathit{iso}}=\{(t_1,\varnothing ),(t_2,\{1,2,\dots ,n,n+1,\dots \})\}.$ Now, ${(H_3,T)}^{\mathit{iso}}\tilde{\cup }{(H_4,T)}^{\mathit{iso}}=\{(t_1,\{1,2,\dots ,n,n+1,\dots \}),(t_2,\{1,2,\dots ,n,n+1,\dots \})\},$ but ${[(H_3,T)\tilde{\cup }(H_4,T)]}^{\mathit{iso}}=\tilde{\mathbf{Z}}.$

6. ${(H_1,T)}^{\mathit{dscl}}={(H_3,T)}^{\mathit{dscl}}=\tilde{\mathbf{Z}},$ but ${[(H_1,T)\tilde{\cap }(H_3,T)]}^{\mathit{dscl}}=\{(t_1,\{\dots ,1,2,3\}),(t_2,\varnothing )\}$

Proposition 3.13.

Let (H, T) be a soft subset of $(A,\mu ,T,⪯)$ and $P_t^a\in \tilde{A}$. Then $P_t^a\in {(H,T)}^{\mathit{iscl}}$ (resp. $P_t^a\in {(H,T)}^{\mathit{dscl}}$) if and only if $(G,T)\tilde{\cap }(H,T)\ne \tilde{\Phi }$ for every decreasing (resp. increasing) supra soft open set (G, T) containing $P_t^a.$

Proof.

$[\Rightarrow ]:$ Let $P_t^a\in {(H,T)}^{\mathit{iscl}}.$ Suppose that there exists a decreasing supra soft open set (G, T) containing $P_t^a$ such that $(G,T)\tilde{\cap }(H,T)=\tilde{\Phi }.$ Then $(H,T)\tilde{\subseteq }(G^c,T).$ So ${(H,T)}^{\mathit{iscl}}\tilde{\subseteq }(G^c,T).$ Thus, $P_t^a\notin {(H,T)}^{\mathit{iscl}}.$ But this contradicts that $P_t^a\in {(H,T)}^{\mathit{iscl}}.$ This shows that the necessary condition holds.

$[\Leftarrow ]:$ Let $(G,T)\tilde{\cap }(H,T)\ne \tilde{\Phi }$ for every decreasing supra soft open set (G, T) containing $P_t^a.$ Suppose that $P_t^a\notin {(H,T)}^{\mathit{iscl}}.$ Then there exists an increasing supra soft closed set (F, T) containing (H, T) such that $P_t^a\notin (F,T).$ So $P_t^a\in (F^c,T)$ and $(F^c,T)\tilde{\cap }(H,T)=\tilde{\Phi }.$ But this contradicts the given condition. This shows that the sufficient condition holds.

A similar proof is given for the case between parentheses. □

Definition 3.14.

Let (H, T) be a soft subset of an SSTOS $(A,\mu ,T,⪯)$ and $P_t^a\in \tilde{A}.$ We say that $P_t^a\in {(H,T)}^{\mathit{isl}}$ (resp. $P_t^a\in {(H,T)}^{\mathit{dsl}}$) provided that $[(G,T)\tilde{\cap }(H,T)]\setminus P_t^a\ne \tilde{\Phi }$ for every decreasing (resp. increasing) supra soft open set (G, T) containing $P_t^a.$

Proposition 3.15.

Let (H, T) and (G, T) be two soft subsets of an SSTOS $(A,\mu ,T,⪯)$. Then the following properties hold.

1. If $(H,T)\tilde{\subseteq }(G,T)$, then ${(H,T)}^{\mathit{isl}}\tilde{\subseteq }{(G,T)}^{\mathit{isl}}$ and ${(H,T)}^{\mathit{dsl}}\tilde{\subseteq }{(G,T)}^{\mathit{dsl}}.$

2. ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{isl}}\tilde{\subseteq }{(H,T)}^{\mathit{isl}}\tilde{\cap }{(G,T)}^{\mathit{isl}}$ and ${[(H,T)\tilde{\cap }(G,T)]}^{\mathit{dsl}}\tilde{\subseteq }{(H,T)}^{\mathit{dsl}}\tilde{\cap }{(G,T)}^{\mathit{dsl}}.$

3. ${(H,T)}^{\mathit{isl}}\tilde{\cup }{(G,T)}^{\mathit{isl}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{isl}}$ and ${(H,T)}^{\mathit{dsl}}\tilde{\cup }{(G,T)}^{\mathit{dsl}}\tilde{\subseteq }{[(H,T)\tilde{\cup }(G,T)]}^{\mathit{dsl}}.$

Proof.

The proof is straightforward. □

Theorem 3.16.

The following properties hold for a soft subset (H, T) of an SSTOS $(A,\mu ,T,⪯).$

1. (H, T) is an increasing supra soft closed set if and only if ${(H,T)}^{\mathit{isl}}\tilde{\subseteq }(H,T).$

2. $(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}$ is an increasing supra soft closed set.

3. ${(H,T)}^{\mathit{iscl}}=(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}.$

Proof.

(i) Necessity: Suppose that (H, T) is an increasing supra soft closed set and let $P_t^a\notin (H,T).$ Then $P_t^a\in {(H,T)}^c$ which is decreasing supra soft open. Because ${(H,T)}^c\tilde{\cap }(H,T)=\tilde{\Phi },$ then $P_t^a\notin {(H,T)}^{\mathit{isl}}.$ Therefore, ${(H,T)}^{\mathit{isl}}\tilde{\subseteq }(H,T).$

Sufficiency: Let $P_t^a\in {(H,T)}^c$ and ${(H,T)}^{\mathit{isl}}\tilde{\subseteq }(H,T).$ Then $P_t^a\notin {(H,T)}^{\mathit{isl}}.$ Therefore, there exists a decreasing supra soft open set $(G_{\mathit{xt}},T)$ such that $[(G_{\mathit{xt}},T)\setminus P_t^a]\tilde{\cap }(H,T)=\tilde{\Phi }.$ Owing to $P_t^a\in {(H,T)}^c,$ then $(G_{\mathit{xt}},T)\tilde{\cap }(H,T)=\tilde{\varnothing }.$ Now, $(G_{\mathit{xt}},T)\tilde{\subseteq }{(H,T)}^c.$ Therefore, ${(H,T)}^c={\cup }_{\forall P_t^a\in {(H,T)}^c}^{˜}(G_{\mathit{xt}},T).$ Thus, ${(H,T)}^c$ is decreasing supra soft open. This finishes the proof.

(ii) Let $P_t^a\notin [(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}].$ Then $P_t^a\notin (H,T)$ and $P_t^a\notin {(H,T)}^{\mathit{isl}}.$ Therefore, there exists a decreasing supra soft open set (G, T) such that (3.1) $$(G,T)\tilde{\cap }(H,T)=\tilde{\Phi }$$(3.1)

Now, for each $P_t^a\in (G,T),$ we have $[(G,T)\setminus P_t^a]\tilde{\cap }(H,T)=\tilde{\Phi },$ so that $P_t^a\notin {(H,T)}^{\mathit{isl}}.$ This automatically implies that (3.2) $$(G,T)\tilde{\cap }{(H,T)}^{\mathit{isl}}=\tilde{\Phi }$$(3.2)

Owing to (3.1)(3.1) $$(G,T)\tilde{\cap }(H,T)=\tilde{\Phi }$$(3.1) and (3.2)(3.2) $$(G,T)\tilde{\cap }{(H,T)}^{\mathit{isl}}=\tilde{\Phi }$$(3.2) above, we obtain $(G,T)\tilde{\cap }[(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}]=\tilde{\Phi }.$ So $P_t^a\notin [(H,T)\cup$ ${(H,T)}^{\mathit{isl}}{]}^{\mathit{isl}}.$ Hence, ${[(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}]}^{\mathit{isl}}\tilde{\subseteq }[(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}].$ It follows from $(\mathbf{i})$ that $(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}$ is increasing supra soft closed.

(iii) Since $(H,T)\tilde{\subseteq }{(H,T)}^{\mathit{iscl}},$ then ${(H,T)}^{\mathit{isl}}\tilde{\subseteq }{[{(H,T)}^{\mathit{iscl}}]}^{\mathit{isl}}.$ Now, ${(H,T)}^{\mathit{iscl}}$ is an increasing supra soft closed set, so that ${[{(H,T)}^{\mathit{iscl}}]}^{\mathit{isl}}\tilde{\subseteq }{(H,T)}^{\mathit{iscl}}.$ This shows that $(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}\tilde{\subseteq }$ ${(H,T)}^{\mathit{iscl}}.$ On the other hand, ${(H,T)}^{\mathit{iscl}}$ is the smallest increasing supra soft closed set containing (H, T). It follows by (ii) above that ${(H,T)}^{\mathit{iscl}}\tilde{\subseteq }(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}.$ Hence, ${(H,T)}^{\mathit{iscl}}=(H,T)\tilde{\cup }{(H,T)}^{\mathit{isl}}.$□

Corollary 3.17.

Let (H, T) be a soft subsets of an SSTOS $(A,\mu ,T,⪯)$. Then the following properties hold.

1. If ${(H,T)}^{\mathit{isl}}=\tilde{\Phi },$ then (H, T) is increasing supra soft closed.

2. If (H, T) is increasing supra soft closed, then ${(H,T)}^{\mathit{isl}}$ is increasing supra soft closed.

Theorem 3.18.

The following properties hold for a soft subset (H, T) of an SSTOS $(A,\mu ,T,⪯).$

1. (H, T) is a decreasing supra soft closed set if and only if ${(H,T)}^{\mathit{dsl}}\tilde{\subseteq }(H,T).$

2. $(H,T)\tilde{\cup }{(H,T)}^{\mathit{dsl}}$ is a decreasing supra soft closed set.

3. ${(H,T)}^{\mathit{dscl}}=(H,T)\tilde{\cup }{(H,T)}^{\mathit{dsl}}.$

Proof.

The proof is similar to that of Theorem (3.16). □

Corollary 3.19.

Let (H, T) be a soft subsets of an SSTOS $(A,\mu ,T,⪯)$. Then the following properties hold.

1. If ${(H,T)}^{\mathit{dsl}}=\tilde{\Phi }$, then (H, T) is decreasing supra soft closed.

2. If (H, T) is decreasing supra soft closed, then ${(H,T)}^{\mathit{dsl}}$ is decreasing supra soft closed.

4. Ordered supra soft separation axioms

We devote this section to introducing ordered supra soft separation axioms, namely supra p-soft T_i-ordered spaces $(i=0,1,2,3,4)$ and to studying their main properties. Various examples are considered to elucidate the relationships between them and to show some obtained results.

Definition 4.1.

An SSTOS $(A,\mu ,T,⪯)$ is said to be:

1. Lower supra p-soft T₁-ordered provided that for every $a\cancel{⪯}b$ in A, there exists an increasing supra soft neighbourhood (W, T) of a such that $b\cancel{⋐}(W,T).$

2. Upper supra p-soft T₁-ordered provided that for every $a\cancel{⪯}b$ in A, there exists a decreasing supra soft neighbourhood (W, T) of b such that $a\cancel{⋐}(W,T).$

3. Supra p-soft T₀-ordered if it is lower supra soft T₁-ordered or upper supra soft T₁-ordered.

4. Supra p-soft T₁-ordered if it is lower supra soft T₁-ordered and upper supra soft T₁-ordered.

5. Supra p-soft T₂-ordered provided that for every $a\cancel{⪯}b$ in A, there exist an increasing supra soft neighbourhood (V, T) of a and a decreasing supra soft neighbourhood (W, T) of b such that $(V,T)\tilde{\cap }(W,T)=\tilde{\Phi }.$

Proposition 4.2.

1. Every supra p-soft T_i-ordered space $(A,\mu ,T,⪯)$ is supra p-soft $T_{i-1}$-ordered for i = 1, 2.

2. Every supra p-soft T_i-ordered space $(A,\mu ,T,⪯)$ is supra p-soft T_i for i = 0, 1, 2.

Proof.

The proof is straightforward. □

To see that converse of the above proposition fails, we present the following two examples.

Example 4.3.

Assume that $(\mathbf{Z},\mu ,T,⪯)$ is the same as in Example (3.12). Then for each $a\ne b\in \mathbf{Z},$ we can find two supra soft open sets (V, T) and (W, T) such that $V(t)=\{a,1\}$ and $W(t)=\{b,2\}$ for each $t\in T.$ Obviously, $a\in (V,T),b\in (W,T)$ and $(V,T)\tilde{\cap }(W,T)=\tilde{\Phi }.$ So that $(\mathbf{Z},\mu ,T,⪯)$ is a supra soft T₂-space. On the other hand, $5\cancel{⪯}3$ and for any increasing supra soft open set (G, T) containing 5, we find that $3⋐(G,T).$ So $(\mathbf{Z},\mu ,T,⪯)$ it is not lower supra p-soft T₀-ordered. Also, $-1\cancel{⪯}-3$ and for any decreasing supra soft open set (G, T) containing –3, we find that $-1⋐(G,T).$ So $(\mathbf{Z},\mu ,T,⪯)$ it is not upper supra p-soft T₀-ordered. Hence, it is not supra p-soft T₀-ordered.

Example 4.4.

Let $T=\{t_1,t_2\}$ be a set of parameters and ${⪯}_1=△\cup \{(a,b),(a,c)\}$ and ${⪯}_2=△\cup \{(a,b)\}$ be two partial order relations on $A=\{a,b,c\}.$ Consider the following soft sets defined as follows: $$\begin{array}{c}(G_1,T)=\{(t_1,\left\{c\right\}),(t_2,\left\{c\right\})\};\\ (G_2,T)=\{(t_1,\{a,b\}),(t_2,\{a,b\})\};\\ (G_3,T)=\{(t_1,\{a,c\}),(t_2,\{a,c\})\};\\ (G_4,T)=\{(t_1,\{b,c\}),(t_2,\{b,c\})\}.\end{array};$$

Then ${\mu }_1=\{\tilde{\Phi },\tilde{A},(G_2,T),(G_3,T),(G_4,T)\}$ and ${\mu }_2=\{\tilde{\Phi },\tilde{A},(G_1,T),(G_2,T),(G_3,T),(G_4,T)\}$ are two supra soft topologies on A. It can be seen that $(A,{\mu }_1,T,{⪯}_1)$ and $(A,{\mu }_2,T,{⪯}_2)$ are supra p-soft T₀-ordered and supra p-soft T₁-ordered spaces, respectively. On the other hand, $b\cancel{⪯}c$ and any increasing supra soft open subset of $(A,{\mu }_1,T,{⪯}_1)$ containing b contains c as well. Hence, $(A,{\mu }_1,T,{⪯}_1)$ is not supra p-soft T₁-ordered. Also, $b⪯/a$ and any increasing supra soft open subset of $(A,{\mu }_2,T,{⪯}_2)$ containing b intersects any decreasing supra soft open subset of $(A,{\mu }_2,T,{⪯}_2)$ containing a. Hence, $(A,{\mu }_2,T,⪯)$ is not supra p-soft T₂-ordered.

Theorem 4.5.

Let $(A,\mu ,T,⪯)$ be an SSTOS. Then the following three statements are equivalent:

1. $(A,\mu ,T,⪯)$ is upper supra p-soft T₁-ordered;

2. For all $a,b\in A$ such that $a⪯/b$, there is a supra soft open set (G, T) containing b such that $a\cancel{⪯}x$ for every $x⋐(G,T)$;

3. $(i(a),T)$ is a supra soft closed set for every $a\in A.$

Proof.

(i) → (ii): Consider $(A,\mu ,T,⪯)$ is an upper supra p-soft T₁-ordered space and let $a\cancel{⪯}b\in A.$ Then there exists a decreasing supra soft neighbourhood (U, T) of b such that $a\cancel{⋐}(U,T).$ Putting $(G,T)={(U,T)}^{\mathit{so}}.$ Suppose that $(G,T)\widetilde{⊈}({(i(a))}^c,T).$ Then there exists $x⋐(G,T)$ and $x\cancel{⋐}({(i(a))}^c,T).$ Therefore, $x\in ((i(a)),T)$ and this implies that $a⪯x.$ Now, $x⋐(U,T)$ implies that $a⋐(U,T).$ But this contradicts that $a\cancel{⋐}(U,T).$ Thus, $(G,T)\tilde{\subseteq }({(i(a))}^c,T).$ Hence, $a\cancel{⪯}x$ for every $x⋐(G,T).$

(ii) → (iii): Consider $a\in A$ and let $x⋐({(i(a))}^c,T).$ Then $a\cancel{⪯}x.$ So there exists a supra soft open set (G, T) containing x such that $(G,T)\tilde{\subseteq }({(i(a))}^c,T).$ Since x and a are chosen arbitrary, then a soft set $({(i(a))}^c,T)$ is supra soft open for every $a\in A.$ Hence, $(i(a),T)$ is a supra soft closed set for all $a\in A.$

(iii) → (i): Let $a\cancel{⪯}b$ in A. Obviously, $(i(a),T)$ is an increasing soft set and by hypothesis, it is supra soft closed as well. Then $({(i(a))}^c,T)$ is a decreasing supra soft open set satisfies that $b\in ({(i(a))}^c,T)$ and $a\cancel{⋐}({(i(a))}^c,T).$ Hence, the desired results are proved. □

Corollary 4.6.

If a is the largest element of an upper supra p-soft T₁-ordered space $(A,\mu ,T,⪯)$, then (a, T) is an increasing supra soft closed set.

Theorem 4.7.

Let $(A,\mu ,T,⪯)$ be an SSTOS. Then the following three statements are equivalent:

1. $(A,\mu ,T,⪯)$ is lower supra p-soft T₁-ordered;

2. For all $a,b\in A$ such that $a\cancel{⪯}b$, there is a supra soft open set (G, T) containing a in which $x\cancel{⪯}b$ for every $x⋐(G,T)$;

3. $(d(a),T)$ is a supra soft closed set for all $a\in X.$

Proof.

The proof is similar to that of Theorem (4.5). □

Corollary 4.8.

If a is the smallest element of a lower supra p-soft T₁-ordered space $(A,\mu ,T,⪯)$, then (a, T) is a decreasing supra soft closed set.

Theorem 4.9.

An SSTOS $(A,\mu ,T,⪯)$ is supra p-soft T₂-ordered if and only if for every $a\cancel{⪯}b$ in A, there exist supra soft open sets (G, T) and (H, T) containing a and b, respectively, such that $x\cancel{⪯}y$ for every $x\in G(t)$ and $y\in H(t).$

Proof.

Necessity: Consider $(A,\mu ,T,⪯)$ is supra p-soft T₂-ordered and let $a,b\in A$ such that $a\cancel{⪯}b.$ Then there exist disjoint supra soft neighbourhoods (V, T) and (W, T) of a and b, respectively, such that (V, T) is increasing and (W, T) is decreasing. Let $x⋐{(V,T)}^{\mathit{so}}=(G,T)$ and $y⋐{(W,T)}^{\mathit{so}}=(H,T).$ Suppose that there exists $t\in T$ such that $x\in G(t),y\in H(t)$ and $x⪯y.$ Owing to (V, T) is increasing and (W, T) is decreasing, we obtain $(V,T)\tilde{\cap }(W,T)\ne \tilde{\Phi },$ which contradicts the disjointness between them. Hence, $x\cancel{⪯}y$ for every $x\in G(t)$ and $y\in H(t).$

Sufficiency: Let $a\cancel{⪯}b$ in A and assume that for every supra soft open sets (G, T) and (H, T) containing a and b, respectively, we have that $i(G,T)\tilde{\cap }d(H,T)\ne \tilde{\Phi }.$ Then there exists $t\in T$ such that $z\in i(G(t))\tilde{\cap }d(H(t)).$ Therefore, there exist $x\in G(t)$ and $y\in H(t)$ such that $x⪯z$ and $z⪯y.$ This means that $x⪯y.$ But this contradicts that $x\cancel{⪯}y$ for every $x\in G(t)$ and $y\in H(t).$ Thus, $i(G,T)\tilde{\cap }d(H,T)=\tilde{\Phi }.$ Owing to i(G, T) is an increasing supra soft neighbourhood of a and d(H, T) is a decreasing supra soft neighborhood of b, the proof is completed. □

Proposition 4.10.

If $(A,\mu ,T,⪯)$ is an SSTOS, then for each $t\in T$, a family ${\mu }_t=\{G(t):(G,T)\in \mu \}$ with a partial order relation $⪯$, form an ordered supra topology on A.

Proof.

Since a family μ_t forms a supra topology on A and $⪯$ is a partial order relation on A, then the triple $(A,{\mu }_t,⪯)$ forms a supra topological ordered space. □

Proposition 4.11.

If an SSTOS $(A,\mu ,T,⪯)$ is supra p-soft T_i-ordered, then a supra topological ordered space $(A,{\mu }_t,⪯)$ is supra T_i-ordered for i = 0, 1, 2.

Proof.

We prove the proposition when i = 2 and the other two cases are proven similarly.

Let $a\cancel{⪯}b$ in $(A,{\mu }_t,⪯)$ in a supra p-soft T₂-ordered space $(A,\mu ,T,⪯).$ Then there exist disjoint an increasing supra soft neighbourhood (V, T) of a and a decreasing supra soft neighbourhood (W, T) of b such that $b\cancel{⋐}(V,T)$ and $a\cancel{⋐}(W,T).$ Therefore, V(t) is an increasing supra neighbourhood of a and W(t) is a decreasing supra neighbourhood of b in $(A,{\mu }_t,⪯)$ such that $V(t)\cap W(t)=\varnothing .$ Thus, a supra topological ordered space $(A,{\mu }_t,⪯)$ is supra T₂-ordered. □

We give the next example to elucidate that the converse of the above proposition fails.

Example 4.12.

Let $T=\{t_1,t_2\}$ be a set of parameters and $⪯=△\cup \{(a,b)\}$ be a partial order relation on $A=\{a,b\}.$ Consider the following soft sets defined as follows: $$(G_1,T)=\{(t_1,\left\{a\right\}),(t_2,A)\};$$ $$(G_2,T)=\{(t_1,A),(t_2,\left\{b\right\})\}\text{\hspace{0.5em}and\hspace{0.5em}}$$ $$(G_3,T)=\{(t_1,\left\{b\right\}),(t_2,\left\{a\right\})\}.$$

Obviously, $\mu =\{\tilde{\Phi },\tilde{A},(G_1,T),(G_2,T),(G_3,T)\}$ is a supra soft topology on A. It can be noted that $(A,\mu ,T,⪯)$ is not a supra p-soft T₀-ordered space. On the other hand, $(A,{\mu }_t,⪯)$ is a supra T₂-ordered space for each $t\in T.$

Definition 4.13.

Let $Y\subseteq A$ and $(A,\mu ,T,⪯)$ be an SSTOS. Then $(Y,{\mu }_Y,T,{⪯}_Y)$ is called supra soft ordered subspace of $(A,\mu ,T,⪯)$ provided that $(Y,{\mu }_Y,T)$ is supra soft subspace of $(A,\mu ,T)$ and ${⪯}_Y=⪯\cap Y\times Y.$

Definition 4.14.

A property is said to be a hereditary property if the property passes from a supra soft topological ordered space to every supra soft ordered subspace.

Theorem 4.15.

The property of being a supra p-soft T_i-ordered space is hereditary for i = 0, 1, 2.

Proof.

Let $(Y,{\mu }_Y,T,{⪯}_Y)$ be a supra soft ordered subspace of a supra p-soft T₂-ordered space $(A,\mu ,T,⪯).$ If $a,b\in Y$ such that $a{\cancel{⪯}}_{Y}b,$ then $a\cancel{⪯}b.$ So by hypothesis, there exist disjoint supra soft neighbourhoods (V, T) and (W, T) of a and b, respectively, such that (V, T) is increasing and (W, T) is decreasing. In $(Y,{\mu }_Y,T,{⪯}_Y),$ we have $(F,T)=\tilde{Y}\tilde{\cap }(V,T)$ and $(G,T)=\tilde{Y}\tilde{\cap }(W,T)$ are disjoint supra soft neighbourhoods of a and b, respectively. In view of Lemma (2.32), we obtain that (F, T) is increasing and (G, T) is decreasing. Hence, $(Y,{\mu }_Y,T,{⪯}_Y)$ is supra p-soft T₂-ordered.

The theorem can be proven similarly in the case of i = 0, 1. □

Definition 4.16.

An SSTOS $(A,\mu ,T,⪯)$ is said to be:

1. Lower (resp. Upper) supra p-soft regularly ordered if for each decreasing (resp. increasing) supra soft closed set (H, T) and $x\in A$ such that $x\cancel{⋐}(H,T),$ there exist disjoint supra soft neighbourhoods (V, T) of (H, T) and (W, T) of x such that (V, T) is decreasing (resp. increasing) and (W, T) is increasing (resp. decreasing).

2. Supra p-soft regularly ordered if it is both lower supra p-soft regularly ordered and upper supra p-soft regularly ordered.

3. Lower (resp. Upper) supra p-soft T₃-ordered if it is both lower (resp. upper) supra p-soft T₁-ordered and lower (resp. upper) supra p-soft regularly ordered.

4. Supra p-soft T₃-ordered if it is both lower supra p-soft T₃-ordered and upper supra p-soft T₃-ordered.

Theorem 4.17.

An SSTOS $(A,\mu ,T,⪯)$ is lower (resp. upper) supra p-soft regularly ordered if and only if for all $x\in X$ and every increasing (resp. decreasing) supra soft open set (U, T) containing x, there is an increasing (resp. a decreasing) supra soft neighbourhood (V, T) of x satisfies that ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }(U,T).$

Proof.

We will start with the proof for lower case, as the proof for the case between parentheses is analogous.

Necessity: Let $x\in A$ and (U, T) be an increasing supra soft open set containing x. Then $(U^c,T)$ is a decreasing supra soft closed set such that $x\cancel{⋐}(U^c,T).$ By hypothesis, there exist disjoint supra soft neighbourhoods (V, T) of x and (W, T) of $(U^c,T)$ such that (V, T) is increasing and (W, T) is decreasing. So there is a supra soft open set (G, T) such that $(U^c,T)\tilde{\subseteq }(G,T)\tilde{\subseteq }(W,T).$ Since $(V,T)\tilde{\subseteq }(W^c,T),$ then $(V,T)\tilde{\subseteq }(G^c,T)\tilde{\subseteq }(U,T)$ and since $(G^c,T)$ is supra soft closed, then ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }(G^c,T)\tilde{\subseteq }(U,T).$

Sufficiency: Let $x\in A$ and (H, T) be a decreasing supra soft closed set such that $x\cancel{⋐}(H,T).$ Then $(H^c,T)$ is an increasing supra soft open set containing x. So that, by hypothesis, there is an increasing supra soft neighbourhood (V, T) of x such that ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }(H^c,T).$ Now, ${({(V,T)}^{\mathit{scl}})}^c=(M,T)$ is a supra soft open set containing (H, T). Thus, d(M, T) is a decreasing supra soft neighbourhood of (H, T). Suppose that $(V,T)\tilde{\cap }d(M,T)\ne \tilde{\Phi }.$ Then there exists $y\in A$ and there exists $t\in T$ such that $y\in V(t)$ and $y\in M(t).$ So there exists $z\in M(t)$ satisfies that $y⪯z.$ This means that $z\in V(t).$ But this contradicts the disjointness between (V, T) and (M, T). Thus, $(V,T)\tilde{\cap }d(M,T)=\tilde{\Phi }.$ This finishes the proof. □

Proposition 4.18.

The following three properties are equivalent if $(A,\mu ,T,⪯)$ is supra p-soft regularly ordered:

1. $(A,\mu ,T,⪯)$ is supra p-soft T₂-ordered;

2. $(A,\mu ,T,⪯)$ is supra p-soft T₁-ordered;

3. $(A,\mu ,T,⪯)$ is supra p-soft T₀-ordered.

Proof.

The direction (i) →(ii) →(iii) is obvious.

To prove that (iii) →(i), let $a,b\in A$ such that $a\cancel{⪯}b.$ Since $(A,\mu ,T,⪯)$ is supra p-soft T₀-ordered, then it is lower supra p-soft T₁-ordered or upper supra p-soft T₁-ordered. Say, it is upper supra p-soft T₁-ordered. From Theorem (4.5), we have $(i(a),T)$ is a supra soft closed set. Obviously, $(i(a),T)$ is increasing and $b\cancel{⋐}(i(a),T).$ Since $(A,\mu ,T,⪯)$ is supra p-soft regularly ordered, then there exist disjoint supra soft neighbourhoods (V, T) and (W, T) of b and $(i(a),T),$ respectively, such that (V, T) is decreasing and (W, T) is increasing. Thus, $(A,\mu ,T,⪯)$ is supra p-soft T₂-ordered. □

Corollary 4.19.

The following three properties are equivalent if $(A,\mu ,T,⪯)$ is upper (resp. lower) supra p-soft regularly ordered:

1. $(A,\mu ,T,⪯)$ is supra p-soft T₂-ordered;

2. $(A,\mu ,T,⪯)$ is supra p-soft T₁-ordered;

3. $(A,\mu ,T,⪯)$ is upper (resp. lower) supra p-soft T₁-ordered.

Definition 4.20.

An SSTOS $(A,\mu ,T,⪯)$ is said to be:

1. Supra soft normally ordered if for each disjoint supra soft closed sets (F, T) and (H, T) such that (F, T) is increasing and (H, T) is decreasing, there exist disjoint supra soft neighbourhoods (V, T) of (F, T) and (W, T) of (H, T) such that (V, T) is increasing and (W, T) is decreasing.

2. Supra p-soft T₄-ordered if it is supra soft normally ordered and supra p-soft T₁-ordered.

Theorem 4.21.

An SSTOS $(A,\mu ,T,⪯)$ is supra soft normally ordered if and only if for every decreasing (resp. increasing) supra soft closed set (F, T) and every decreasing (resp. increasing) supra soft open neighbourhood (U, T) of (F, T), there is a decreasing (resp. an increasing) supra soft neighbourhood (V, T) of (F, T) satisfies that ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }(U,T).$

Proof.

We prove the theorem in the case of decreasing and the increasing case can be proven similarly.

Necessity: Let (F, T) be a decreasing supra soft closed set and (U, T) be a decreasing supra soft open neighbourhood of (F, T). Then $(U^c,T)$ is an increasing supra soft closed set and $(F,T)\tilde{\cap }(U^c,T)=\tilde{\Phi }.$ Since $(A,\mu ,T,⪯)$ is supra soft normally ordered, then there exist a decreasing supra soft neighbourhood (V, T) of (F, T) and an increasing supra soft neighbourhood (W, T) of $(U^c,T)$ such that $(V,T)\tilde{\cap }(W,T)=\tilde{\Phi }.$ Since (W, T) is a supra soft neighbourhood of $(U^c,T),$ then there exists a supra soft open set (H, T) such that $(U^c,T)\tilde{\subseteq }(H,T)\tilde{\subseteq }(W,T).$ Now, $(W^c,T)\tilde{\subseteq }(H^c,T)\tilde{\subseteq }(U,T)$ and $(V,T)\tilde{\subseteq }(W^c,T).$ So it follows that ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }{(W^c,T)}^{\mathit{scl}}\tilde{\subseteq }(H^c,T)\tilde{\subseteq }(U,T).$ Thus, the necessary part holds.

Sufficiency: Let $(F_1,T)$ and $(F_2,T)$ be two disjoint supra soft closed sets such that $(F_1,T)$ is decreasing and $(F_2,T)$ is increasing. Then $(F_2^c,T)$ is a decreasing supra soft open set containing $(F_1,T).$ By hypothesise, there exists a decreasing supra soft neighbourhood (V, T) of $(F_1,T)$ such that ${(V,T)}^{\mathit{scl}}\tilde{\subseteq }(F_2^c,T).$ Setting $(H,T)=\tilde{A}\setminus {(V,T)}^{\mathit{scl}}.$ This means that (H, T) is a supra soft open set containing $(F_2,T).$ Obviously, $(F_2,T)\tilde{\subseteq }(H,T),(F_1,T)\tilde{\subseteq }(V,T)$ and $(H,T)\tilde{\cap }(V,T)=\tilde{\Phi }.$ Now, i(H, T) is an increasing supra soft neighbourhood of $(F_2,T).$ Suppose that $i(H,T)\tilde{\cap }(V,T)\ne \tilde{\Phi }.$ Then there exists $t\in T$ such that $x\in i(H(t))$ and $x\in V(t)=d(V(t)).$ This implies that there exist $a\in H(t)$ and $b\in V(t)$ such that $a⪯x$ and $x⪯b.$ Owing to $⪯$ is transitive, we obtain $a⪯b.$ Therefore, $b⋐(H,T)\tilde{\cap }(V,T).$ This contradicts the disjointness between (H, T) and (V, T). Thus, $i(H,T)\tilde{\cap }(V,T)=\tilde{\Phi }.$ Hence, the proof is completed. □

Proposition 4.22.

1. Every supra p-soft T_i-ordered space $(A,\mu ,T,⪯)$ is supra p-soft $T_{i-1}$-ordered for i = 3, 4.

2. Every supra p-soft T_i-ordered space $(A,\mu ,T,⪯)$ is supra p-soft T_i for i = 3, 4.

Proof.

(i) From Proposition (4.18), we obtain that every supra p-soft T₃-ordered space is supra p-soft T₂-ordered.

To prove the proposition in the case of i = 4, let $a\in A$ and (F, T) be a decreasing supra soft closed set such that $a\cancel{⋐}(F,T).$ Since $(A,\mu ,T,⪯)$ is supra p-soft T₁-ordered, then $(i(a),T)$ is an increasing supra soft closed set and since $(A,\mu ,T,⪯)$ is supra soft normally ordered, then there exist disjoint supra soft neighbourhoods (V, T) and (W, T) of $(i(a),T)$ and (F, T), respectively, such that (V, T) is increasing and (W, T) is decreasing. Therefore, $(A,\mu ,T,⪯)$ is lower supra p-soft regularly ordered. If (F, T) is an increasing soft closed set, then we similarly prove that $(A,\mu ,T,⪯)$ is upper supra p-soft regularly ordered. Thus, $(A,\mu ,T,⪯)$ is supra p-soft regularly ordered. Hence, $(A,\mu ,T,⪯)$ is supra p-soft T₃-ordered.

(ii) The proof is straightforward.□

The converse of the above proposition is not always true as illustrated in the following two examples.

Example 4.23.

We consider $T=\left\{t\right\}$ and $⪯$ is an equality relation. Then the concepts of supra soft topological ordered spaces and supra topological spaces are identical. So Example 1 and Example 6 in (Al-shami, 2016) showed that the converse of item (i) above is not true.

Example 4.24.

It is well known that a soft topology is a supra soft topology, so that we suffice with Example 4.28 and Example 4.29 in (Al-shami et al. 2018a) to show that the converse of item (ii) above is not true.

Definition 4.25.

Let $\{(A_i,{\mu }_i,T_i,{⪯}_i):i\in \{1,2,\dots ,n\left\}\right\}$ be a finite family of supra soft topological ordered spaces. The product of these supra soft topological ordered spaces is given by $A={\prod }_{i=1}^n{A}_i,$ μ is the product supra topology on A, $T={\prod }_{i=1}^n{T}_i$ and $⪯=\{(a,b):a,b\in X$ such that $(a_i,b_i)\in {⪯}_i$ for every $i\in \{1,2,\dots ,n\}\},$ where $a=(a_1,a_2,\dots ,a_n)$ and $b=(b_1,b_2,\dots ,b_n).$

Lemma 4.26.

If $(H,T_1\times T_2)$ is a decreasing (resp. an increasing) supra soft closed subset of a supra soft ordered product space $(A\times B,{\mu }_1\times {\mu }_2,T_1\times T_2,⪯)$, then $(H,T_1\times T_2)=[{(G,T_1)}^c\times \tilde{B}]\tilde{\cup }[\tilde{A}\times {(F,T_2)}^c]$ for some increasing (resp. decreasing) supra soft open sets $(G,T_1)\in {\mu }_1$ and $(F,T_2)\in {\mu }_2.$

Proof.

Suppose that $(H,T_1\times T_2)$ is a decreasing supra soft closed subset of a supra soft product space $(A\times B,{\mu }_1\times {\mu }_2,T_1\times T_2,⪯).$ Then from Lemma (2.31), there exist supra soft open sets $(G,T_1)\in {\mu }_1$ and $(F,T_2)\in {\mu }_2$ such that $(H,T_1\times T_2)=[{(G,T_1)}^c\times \tilde{B}]\tilde{\cup }[\tilde{A}\times {(F,T_2)}^c].$ To prove that $(G,T_1)$ and $(F,T_2)$ are increasing, consider that at least one of them is not increasing. Without loss of generality, consider that $(G,T_1)$ is not increasing. Then, ${(G,T_1)}^c$ is not decreasing. It follows that there exist $t\in T_1$ and $a\in A$ such that $P_t^a\in d(G^c,T_1)$ and $P_t^a\notin (G^c,T_1).$ By choosing $P_k^b\notin (F^c,T_2),$ we obtain that $P_{(t,k)}^{(a,b)}\in d[{(G,T_1)}^c\times \tilde{B}]$ and $P_{(t,k)}^{(a,b)}\notin [{(G,T_1)}^c\times \tilde{B}]\tilde{\cup }[\tilde{A}\times {(F,T_2)}^c].$ This implies that $(H,T_1\times T_2)$ is not a decreasing soft set. But this contradicts the given condition. Hence, $(G,T_1)$ and $(F,T_2)$ are increasing soft sets.

A similar proof is given for the case between parentheses. □

Theorem 4.27.

The finite product of supra p-soft T_i-ordered spaces is supra p-soft T_i-ordered for $i=0,1,2,3.$

Proof.

We prove the theorem in the case of i = 2, 3. The other cases follow similar lines.

(i) Consider $(A\times B,\mu ,T,⪯)$ is the product of two supra p-soft T₂-ordered spaces $(A,{\mu }_1,T_1,{⪯}_1)$ and $(B,{\mu }_2,T_2,{⪯}_2)$ and let $(a_1,b_1)\cancel{⪯}(a_2,b_2)\in A\times B.$ Then $a_1⪯{/}_1{a}_2$ or $b_1{\cancel{⪯}}_2{b}_2.$ Without loss of generality, say $a_1{\cancel{⪯}}_1{a}_2.$ Since $(A,{\mu }_1,T_1,{⪯}_1)$ is supra p-soft T₂-ordered, then there exist disjoint supra soft neighbourhoods $(V,T_1)$ and $(W,T_1)$ of a₁ and a₂, respectively, such that $(V,T_1)$ is increasing and $(W,T_1)$ is decreasing. Therefore, $(V,T_1)\times \tilde{B}$ and $(W,T_1)\times \tilde{B}$ are supra soft neighbourhoods of (a₁, b₁) and (a₂, b₂), respectively, such that $[(V,T_1)\times \tilde{B}]\tilde{\cap }[(W,T_1)\times \tilde{B}]={\tilde{\Phi }}_{T_1\times T_2}.$ In view of Theorem (2.20), we obtain $(V,T_1)\times \tilde{B}$ is increasing and $(W,T_1)\times \tilde{B}$ is decreasing, which finishes the proof.

(ii) Consider $(A\times B,\mu ,T,⪯)$ is the product of two supra p-soft T₃-ordered spaces $(A,{\mu }_1,T_1,{⪯}_1)$ and $(B,{\mu }_2,T_2,{⪯}_2)$ and let $(H,T_1\times T_2)$ be a decreasing supra soft closed set. It follows by Lemma (2.31) that $(H,T_1\times T_2)=[{(G,T_1)}^c\times \tilde{B})\tilde{\cup }(\tilde{A}\times {(U,T_2)}^c]$ for some increasing supra soft open sets $(G,T_1)\in {\mu }_1$ and $(U,T_2)\in {\mu }_2.$ For every $(a,b)\cancel{⋐}(H,T_1\times T_2),$ we have $(a,b)\cancel{⋐}{(G,T_1)}^c\times \tilde{B}$ and $(a,b)\cancel{⋐}\tilde{A}\times {(U,T_2)}^c.$ Automatically, it follows that $a\cancel{⋐}{(G,T_1)}^c$ and $b\cancel{⋐}{(U,T_1)}^c.$ Since $(A,{\mu }_1,T_1,{⪯}_1)$ and $(B,{\mu }_2,T_2,{⪯}_2)$ are supra p-soft regularly ordered, then there exist disjoint supra soft neighbourhoods (F₁, T₁) and (F₂, T₁) of a and ${(G,T_1)}^c,$ respectively, such that (F₁, T₁) is increasing and (F₂, T₁) is decreasing, and there exist disjoint supra soft neighbourhood (F₃, T₂) and (F₄, T₂) of b and ${(U,T_2)}^c,$ respectively, such that (F₃, T₂) is increasing and (F₄, T₂) is decreasing. In $(A\times B,\mu ,T,⪯),$ we obtain $[(F_2,T_1)\times \tilde{B})\tilde{\cup }(\tilde{A}\times (F_4,T_2)]$ is a decreasing supra soft neighbourhood of $(H,T_1\times T_2)$ and $(F_1,T_1)\times (F_3,T_2)$ is an increasing supra soft neighbourhood of (a, b). Since $[(F_1,T_1)\times (F_3,T_2)]\tilde{\cap }[(F_2,T_1)\times \tilde{B})\tilde{\cup }(\tilde{A}\times (F_2,T_4))]={\tilde{\Phi }}_{T_1\times T_2},$ then $(A\times B,\mu ,T,⪯)$ is lower supra p-soft regularly ordered. Similarly, one can prove that $(A\times B,\mu ,T,⪯)$ is upper supra p-soft regularly ordered. Hence, $(A\times B,\mu ,T,⪯)$ is supra p-soft T₃-ordered.□

Remark 4.28.

An SSTOS $(A,\mu ,T,⪯)$ is a supra topological space if T is a singleton and $⪯$ is an equality relation. Sorgenfrey Line space, in classical topology, is an example to demonstrate excluding of supra soft T₄-ordered spaces from the result above.

Definition 4.29.

A supra soft ordered subspace $(Y,{\mu }_Y,T,{⪯}_Y)$ of an SSTOS $(A,\mu ,T,⪯)$ is called supra soft compatibly ordered provided that for each increasing (resp. decreasing) supra soft closed subset (H, T) of $(Y,{\mu }_Y,T,{⪯}_Y),$ there exists an increasing (resp. a decreasing) supra soft closed subset $(H^{\star },T)$ of $(A,\mu ,T,⪯)$ such that $(H,T)=\tilde{Y}\tilde{\cap }(H^{\star },T).$

Theorem 4.30.

Every supra soft compatibly ordered subspace $(Y,{\mu }_Y,T,{⪯}_Y)$ of a supra p-soft regularly ordered space $(A,\mu ,T,⪯)$ is supra p-soft regularly ordered.

Proof.

Let $y\in Y$ and (H, T) be a decreasing supra soft closed subset of $(Y,{\mu }_Y,T,{⪯}_Y)$ such that $y\cancel{⋐}(H,T).$ Because the supra soft ordered subspace $(Y,{\mu }_Y,T,{⪯}_Y)$ of $(A,\mu ,T,⪯)$ is supra soft compatibly ordered, there exists a decreasing supra soft closed subset $(H^{\star },T)$ of $(A,\mu ,T,⪯)$ such that $(H,T)=\tilde{Y}\tilde{\cap }(H^{\star },T).$ So that by hypothesis, there exist disjoint supra soft neighbourhoods (V, T) and (W, T) of y and $(H^{\star },T),$ respectively, such that (V, T) is increasing and (W, T) is decreasing. In $(Y,{\mu }_Y,T,{⪯}_Y),$ we get $\tilde{Y}\tilde{\cap }(V,T)$ and $\tilde{Y}\tilde{\cap }(W,T)$ are supra soft neighbourhoods of y and (H, T), respectively. It follows by Lemma (2.32) that $\tilde{Y}\tilde{\cap }(V,T)$ is increasing and $\tilde{Y}\tilde{\cap }(W,T)$ is decreasing. Consequently, $(Y,{\mu }_Y,T,{⪯}_Y)$ is lower supra p-soft regularly ordered.

Similarly, one can prove that $(Y,{\mu }_Y,T,{⪯}_Y)$ is upper supra p-soft regularly ordered. Hence, the proof is completed. □

Corollary 4.31.

Every supra soft compatibly ordered subspace $(Y,{\mu }_Y,T,{⪯}_Y)$ of a supra p-soft T₃-ordered space $(A,\mu ,T,⪯)$ is supra p-soft T₃-ordered.

The proof of the next proposition is routine and so omitted.

Proposition 4.32.

Every supra soft closed compatibly ordered subspace $(Y,{\mu }_Y,T,{⪯}_Y)$ of a supra p-soft T₄-ordered space $(A,\mu ,T,⪯)$ is supra p-soft T₄-ordered.

Definition 4.33.

A property is said to be a supra soft ordered topological property if the property is preserved by an ordered embedding soft $S^{\star }$-homeomorphism mappings.

Lemma 4.34.

Let (G, T) and (H, T) be two disjoint soft sets over A. If $a\in (G,T)$, then $a\cancel{⋐}(H,T).$

Theorem 4.35.

The property of being a supra p-soft T_i-ordered space is a supra soft topological ordered property for $i=0,1,2,3,4.$

Proof.

We prove the theorem in the case of i = 4, and the other follow similar lines.

Suppose that $f_{\varphi }$ is an ordered embedding soft $S^{\star }$-homeomorphism mapping of a supra p-soft T₄-ordered space $(A,\mu ,T,{⪯}_1)$ onto an SSTOS $(B,\theta ,S,{⪯}_2)$ and let $x,y\in B$ such that $x{\cancel{⪯}}_{2}y.$ Then $P_{\beta }^x{\cancel{⪯}}_2{P}_{\beta }^y$ for each $\beta \in S.$ Since $f_{\varphi }$ is bijective, then there exist $P_{\alpha }^a$ and $P_{\alpha }^b$ in $\tilde{A}$ such that $f_{\varphi }(P_{\alpha }^a)=P_{\beta }^x$ and $f_{\varphi }(P_{\alpha }^b)=P_{\beta }^y$ and since $f_{\varphi }$ is an ordered embedding, then $P_{\alpha }^a{\cancel{⪯}}_1{P}_{\alpha }^b.$ So $a{\cancel{⪯}}_{1}b.$ By hypothesis, there exist supra soft neighbourhoods (V, T) and (W, T) of a and b, respectively, such that (V, T) is increasing and (W, T) is decreasing. Since $f_{\varphi }$ is bijective supra soft open, then $f_{\varphi }(V,T)$ and $f_{\varphi }(W,T)$ are disjoint supra soft neighbourhoods of x and y, respectively. It follows, by Theorem (2.22), that $f_{\varphi }(V,T)$ is increasing and $f_{\varphi }(W,T)$ is decreasing.

It remains to prove the soft ordered normality. Let (H, S) and (F, S) be two disjoint supra soft closed sets such that (H, S) is increasing and (F, S) is decreasing. Since $f_{\varphi }$ is bijective soft $S^{\star }$-continuous, then $f_{\varphi }^{-1}(H,S)$ and $f_{\varphi }^{-1}(F,S)$ are disjoint supra soft closed sets and since $f_{\varphi }$ is ordered embedding, then $f_{\varphi }^{-1}(H,S)$ is increasing and $f_{\varphi }^{-1}(F,S)$ is decreasing. By hypothesis, there exist disjoint supra soft neighbourhoods (V, S) and (W, S) of $f_{\varphi }^{-1}(H,S)$ and $f_{\varphi }^{-1}(F,S),$ respectively, such that (V, S) is increasing and (W, S) is decreasing. So $(H,S)\tilde{\subseteq }{f}_{\varphi }(V,S)$ and $(F,S)\tilde{\subseteq }{f}_{\varphi }(W,S).$ The disjointness of the supra soft neighbourhoods $f_{\varphi }(V,S)$ and $f_{\varphi }(W,S)$ finishes the proof. □

5. Conclusion and future work

In this work, we first introduce a concept of supra soft topological spaces and investigate some properties. Then we define and discuss the concepts of supra p-soft T_i-ordered spaces $(i=0,1,2,3,4).$ With the help of examples, we illustrate the relationships between these spaces. We characterize the concepts of supra p-soft T_i-ordered spaces (i = 1, 2), supra p-soft regular ordered and supra p-soft normally ordered spaces. In the forthcoming works, we plan to do the following:

1. Formulating new types of supra soft separation axioms by using $(\in ,\notin ),(⋐,\notin )$ and $(⋐,\cancel{⋐})$ relations and making a comparative study among them.

2. Investigating the possibility of applications of supra soft topological spaces to solve some problems in digital line (Kozae, Shokry, & Zidan, 2016).

3. Utilizing soft somewhere dense sets (Al-shami, 2018) to study new types of soft ordered axioms

Finally, we hope that our findings help the researchers to enhance and promote their studies on soft ordered spaces to carry out a general framework for their applications in life.

Acknowledgements

The authors would like to thank the editors and the referees for their valuable comments which helped us improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.